It is well-known that the existence of traveling
wave solutions for reaction-diffusion partial differential equations can be proved
by showing the existence of certain heteroclinic orbits for related autonomous
planar differential equations. We introduce a method for finding explicit upper
and lower bounds of these heteroclinic orbits. In particular, for the classical
Fisher-Kolmogorov equation we give rational upper and lower bounds which allow to
locate these solutions analytically and with very high accuracy. These results
allow one to construct analytical approximate expressions for the traveling wave
solutions with a rigorous control of the errors for arbitrary values of the
independent variables. These explicit expressions are very simple and tractable
for practical purposes. They are constructed with exponential and rational
functions.